Competitive algorithms for layered graph traversal

A layered graph is a connected graph whose vertices are partitioned into sets L₀ = {s}, L₁, L₂, ..., and whose edges, which have nonnegative integral weights, run between consecutive layers. Its width is max{|L_i|}. In the on-line layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i-1 and i are only revealed when the searcher reaches layer i-1. We give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. We give a deterministic on-line algorithm which is O(9^w)-competitive on width-w graphs and prove that for no w can a deterministic on-line algorithm have a competitive ratio better than 2^w-2 on width-w graphs. We prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized on-line layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, we give a randomized on-line algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor.